Method and apparatus for X-ray or γ-ray 3-D tomography using a fan beam

ABSTRACT

A fan-shaped beam of penetrating radiation, such as X-ray or γ-ray radiation, is directed through a slice of the body to be analyzed to a position sensitive detector for deriving a shadowgraph of transmission or absorption of the penetrating radiation by the body. A number of such shadowgraphs are obtained for different angles of rotation of the fan-shaped beam relative to the center of the slice being analyzed. The detected fan beam shadowgraph data is reordered into shadowgraph data corresponding to sets of parallel paths of radiation through the body. The reordered parallel path shadowgraph data is then convoluted in accordance with a 3-D reconstruction method by convolution in a computer to derive a 3-D reconstructed tomograph of the body under analysis. In a preferred embodiment, the position sensitive detector comprises a multiwire detector wherein the wires are arrayed parallel to the direction of the divergent penetrating rays to be detected. A focussed grid collimator is interposed between the body and the position sensitive detector for collimating the penetrating rays to be detected. The source of penetrating radiation is preferably a monochromatic source.

GOVERNMENT CONTRACT

The Government has rights in this invention pursuant to grant numberGI-35007 awarded by the National Science Foundation.

RELATED CASES

A multiwire radiation detector, of the type wherein the wires of thedetector are parallel to the divergent rays of penetrating radiation,forms the subject matter of and is claimed in copending U.S. applicationSer. No. 528,025 filed Nov. 29, 1974 and assigned to the same assigneeas the present invention. The method and apparatus for 3-D X- or γ-raytomography employing a fan-shaped beam forms the subject matter of andis claimed in copending U.S. application Ser. No. 528,026 filed Nov. 29,1974 and assigned to the same assignee as the present invention.

BACKGROUND OF THE INVENTION

The present invention relates in general to fan beam X- or γ-ray 3-Dtomography and more particularly to such tomography utilizing a positionsensitive detector.

DESCRIPTION OF THE PRIOR ART

Heretofore, it has been proposed to employ collimated beams ofpenetrating radiation to derive a set of angularly displaced shadowgraphdata from which to reconstruct a 3-D tomograph of a slice of the body.The 3-D tomograph was reconstructed by a method of computing theabsorption or transmission coefficients for a matrix of elements ofcross sectional area intersected by the angularly displaced sets ofparallel rays. The coefficients were refined by a process of successiveapproximations to derive the final 3-D tomograph. Such a method isproposed in U.S. Pat. No. 3,778,614 issued Dec. 11, 1973.

In this prior patent, the shadowgraph data is derived by either of twomethods. In a first method, a collimated source of penetrating radiationpasses through the body to a detector in alignment with the beam path.The detector and source are then rectilinearly translated laterally ofthe body to derive a given set of shadowgraph data. The source anddetector are then angularly rotated to a second position and againlaterally translated relative to the body to obtain a second set ofshadowgraph data, and so forth.

In the second method, a fan-shaped array of collimated beams ofpenetrating radiation, each beam having a detector in alignmenttherewith, is caused to be laterally rectilinearly translated relativeto the body and then rotated to a second position with lateraltranslation at the second position, and so forth and so on, to deriveangularly displaced sets of shadowgraph data.

The advantage of the second scheme relative to the first scheme, is thatthe lateral translation can be cut by a factor of 1/N where N is thenumber of detectors, such as 6 or 7. However, this prior art patentdiscloses that the paths of penetrating radiation through the bodyshould all have a constant width and that this is an essentialrequirement for accurate computer calculations which are to follow forreconstruction of the 3-D tomograph. Also, the algorithms presentedtherein for reconstruction of the 3-D tomograph are based upon sets ofparallel rays. However, in the case of the collimated divergent beams,there is no disclosure of how one obtains shadowgraph data based uponsets of parallel rays. Furthermore, there is no teaching nor suggestionof how the divergent rays passing through the body could be made totraverse paths of constant width. Thus, there is no teaching in thesubject patent of a method for reconstruction of 3-D tomographs fromsets of divergent rays of penetration as would be obtained from adivergent fan beam.

It has also been proposed in the prior art relating to 3-D X-ray orγ-ray tomographic reconstructions to reorder divergent fan beamshadowgraphic data into parallel ray shadowgraphic data from which tocompute the 3-D reconstruction. Such a proposal is found in an articletitled, "Reconstruction of Substance From Shadow" appearing in theProceedings of the Indian Academy of Sciences, Vol. LXXIV, No. 1, Sec. A(1971) pages 14-24.

The problem with this reconstruction proposal is that it provides only arather abstract algorithm for transforming a continuous distribtuion ofdivergent X-ray or γ-ray shadowgraphic data into a parallel raycontinuous distribution of X-ray or γ-ray shadowgraphic data. However,in a practical system, the data is acquired not as a continuousdistribution but as incremental data not only as a function of distanceX perpendicular to the central ray of the fan beam but also as afunction of θ, the angle of rotation of the source about the axis ofrevolution. There is no teaching in this article of a way to transformsuch incremented divergent fan beam shadowgraphic data intocorresponding sets of incremented parallel ray shadowgraphic data. Thereis, however, a mention of a required interpolation which is not definedin the article (see page 22 therein).

SUMMARY OF THE PRESENT INVENTION

The principal object of the present invention is the provision of animproved method and apparatus for reconstruction 3-D X- or γ-raytomographs derived from sets of shadowgraphic data derived employing afan-shaped beam of penetrating radiation.

In one feature of the present invention, a fan-shaped beam divergent rayof penetrating radiation is angularly moved relative to the body toobtain angularly displaced shadowgraph data and wherein the shadowgraphdata is reordered to derive shadowgram data based upon angularlydisplaced sets of parallel rays, whereby reconstruction of a 3-Dtomograph is facilitated.

In another feature of the present invention, the 3-D X- or γ-raytomograph is reconstructed from sets of shadowgraphic data derived byuse of a fan-shaped beam of penetrating radiation, such reconstructionbeing by a method of convolutions.

In another feature of the present invention, the divergent penetratingradiation detected is detected at angular spacing less than andpreferably one half the angular spacing between adjacent shadowgrams.

In another feature of the present invention, the divergent ray data isconverted into parallel ray data of equal lateral spacing.

Other features and advantages of the present invention will becomeapparent upon a perusal of the following specification taken inconnection with the accompanying drawings wherein:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic transverse sectional diagram of a penetrating ray3-D tomography apparatus of the present invention and including ashadowgraph produced by the apparatus,

FIG. 2 is an enlarged sectional view of a portion of the structure ofFIG. 1 delineated by line 2--2,

FIG. 3 is an enlarged detail view of a portion of the structure of FIG.1 delineated by line 3--3,

FIG. 4 is a view of the structure of FIG. 3 taken along line 4--4 in thedirection of the arrows,

FIG. 5 is an enlarged detailed view of a portion of the structure ofFIG. 1 taken along line 5--5 in the direction of the arrows,

FIG. 6 is a schematic diagram of a fan beam X-ray 3-D tomographicapparatus incorporating features of the present invention,

FIG. 7 is a longitudinal sectional view of a position sensitive X-raydetector employed in the apparatus of the present invention,

FIG. 8 is a view of the structure of FIG. 7 taken along line 8--8 in thedirection of the arrow and including associated circuitry in blockdiagram form,

FIG. 9 is a schematic line diagram, partly in block diagram form, of adata processing portion of the apparatus of the present invention,

FIG. 10 is a schematic line diagram depicting how a rotating fan beamproduces sets of parallel rays,

FIG. 11 is a view similar to that of FIG. 10 depicting extrapolation ofthe arrangement of FIG. 10 to twice as many detectors and toapproximations of parallelism for the added intermediate rays,

FIG. 12 is a schematic diagram depicting the process for correction ofthe set of detected parallel rays to sets of parallel ray data of equallateral spacing,

FIG. 13 is a schematic diagram depicting the process for compensatingfor non-equal spacing between the detected parallel rays,

FIG. 14 is a shadowgraph converted to 1n of the ratio of detectedintensity as a function of lateral position I'(y) divided by the beamintensity Io measured without absorption,

FIG. 15 is a plot of a function utilized in the 3-D reconstructionmethod,

FIG. 16 is a plot for the convolution of the function of FIG. 15 with asingle point on the function of FIG. 14,

FIG. 17 is a plot of the convolution of the function of FIG. 15 with theshadowgraph function of FIG. 14,

FIG. 18 is a schematic line diagram depicting the process for backprojecting and adding the contributions of the convoluted shadowgraphicdata,

FIG. 19 is a schematic line diagram representing the positionaluncertainty when a fan beam is detected by a rectilinear detectingarray,

FIG. 20 is a longitudinal sectional view of a preferred multiwireradiation detector,

FIG. 21 is an enlarged sectional view of the structure of FIG. 20 takenalong line 21-21 in the direction of the arrows, and

FIG. 22 is a flow chart for a computerized method for reconstruction ofthe 3-D tomographs.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring now to FIG. 1, there is shown an apparatus for derivingpenetrating radiation shadowgraphs of a body to be examined. Moreparticularly, the patient 11 to be examined is supported on a couch 12,as of a suitable plastic material. A suitable point source ofpenetrating radiation 13, such as X-rays or γ-rays is disposed above thebody for projecting a fan-shaped beam of divergent penetrating radiationthrough a narrow elongated slot 14 in a collimator 15, as of lead. Thefan-shaped beam is relatively thin and comprises divergent rays ofpenetrating radiation which are directed onto the body 11 to beexamined.

The penetrating radiation is partially absorbed in the body 11 inaccordance with the density of the various portions of the bodypenetrated by the radiation. In a typical example of a torso tomograph,the lungs would have relatively low density, whereas the spinal columnwould have relatively high density. The penetrating radiation emergingfrom the body is passed through a second fan beam collimator 20 andthence through a focussed grid collimator 16 which is shown in greaterdetail in FIGS. 3 and 4. The second collimator 20 is similar to the fansource collimator 15 and the focussed grid collimator 16 comprises anarray of lead vanes 17 embedded in a plastic filler material 18, as ofpolyethylene. The vanes 17 have a thickness of, for example, 0.5millimeters and the plane of the vanes is directed parallel to thedivergent rays emanating from the source 13. In a typical example, thecollimating vanes 17 are spaced apart by approximately 5.0 millimetersfor blocking scattered radiation emerging from the body 11 from passinginto a position sensitive radiation detector 21. Less than 1% of thescattered radiation reaches the detector 21.

In a preferred embodiment, the position sensitive detector 21 includesan array of closely spaced detecting wires as more fully disclosed belowwith regard to FIGS. 7, 8, 9, and 20 and 21. Generally, there is onedetecting element of the array in alignment with the center of each ofthe bins of collimated divergent rays passing through the collimator 16.In a typical example, the position sensitive detector 21 would have alength of approximately 50 centimeters and would include 150 individualdetecting elements at 1/2° intervals. The fan beam typically subtends anarc ψ of 75°.

The penetrating radiation such as X-rays or γ-rays, in passing throughthe body 11, are variously attenuated or absorbed by the differentportions within the body such as the lungs, spinal column, etc. toproduce a shadowgraph of detected intensity versus distance as shown inFIG. 1 by curve 22.

In a typical example, the X-ray or γ-ray source 13, which is shown ingreater detail in FIG. 2 comprises a cylindrical body 23 of high atomicnumber Z material, such as lead or tantalum, and includes a centralre-entrant bore 24 containing a cylindrical insert 25. The inserttypically comprises a plastic body 26 having a pellet 27 of radioactivematerial embedded in the outer end thereof. A shutter 28 of high Zmaterial is pivotably mounted to the body 23 at 29 and is pivoted forclosing off the source of radiation and held in the closed position viaa spring clasp 31. Typical source materials for the pellet 27 includematerials that will provide X-ray or γ-ray radiation having intensitiesfalling within the range of 50-100 keV. The source radiation ispreferably monochromatic. Materials of this type include Gadolinium¹⁵³having a half life of approximately 242 days and having a very stablepredictable decay rate. However, other types of sources 13 could beemployed such as an X-ray tube utilizing various different types ofsecondary γ- or X-ray emitting materials.

Referring now to FIG. 6 there is shown the apparatus of FIG. 1 mountedfor rotation about an axis of rotation 33 disposed centrally of the body11. The source 13, detector 21, and collimator 16 are mounted to a ring34 for rotation about the axis of rotation 33. The ring 34 is drivenfrom a friction drive wheel 35 which is connected to a drive motor 36via a suitable drive means, such as a drive belt 37. The ring 34 issupported via the drive wheel and an idler wheel 38 rotationally mountedto a base support structure 39. The ring 34 includes an array of axiallydirected pins 41 disposed at one degree intervals around the peripheryof the ring 34. A photocell detector 42 is mounted in fixed relationrelative to the ring 34 and pins 41 so that as the ring 34 is rotatedsuccessive pins 41 come into registration with the optical path ofreflected light from the respective pin 42 to the photocell 42 forgiving an output signal indicative of the angular position of the ring34 and, thus, the source and detector relative to the body 11.

This electrical signal, representative of the position of the ring 34,is fed to one input of a sequencer 40. The output of the sequencer 40 isfed to the motor 36 for driving the ring 34 around the body 11. For each1° of angular position θ, a 151 point shadowgraph is derived so that aset of shadowgraphs is obtained there being one shadowgraph for eachdegree of rotation of the source around the patient. In a typicalexample employing a 75° fan beam of radiation, the sequencer 40 is setto rotate the source continuously around the patient for a total of 255°to obtain 255 sets of shadowgraph data. The reason for 255 sets of datais explained below.

Referring now to FIGS. 7 and 8 there is shown a position sensitivedetector 21. The detector includes an elongated channel member 45, as ofG-10 fiberglass, having a base portion 46 and two upstanding side wallportions 47 and 48. The channel 45 is closed at its ends via transversewalls 49 and 51. In a typical example, the detector 21 has a length of50 centimeters. A penetrating ray transparent gas-tight window 52, as ofMylar, is sealed across the open side of the channel 45. An array oftransversely directed anode wires 53 extend for the length of thedetector 21. Two arrays of longitudinally directed cathode wires 54 and55 are disposed on opposite sides of the anode array 53.

In a typical example, the anode wires 53 are spaced apart by 2.5millimeters and the wires have a diameter of 0.025 millimeters (25microns). The cathode wires 54 and 55 are of tungsten having a diameterof 0.1 millimeters and are spaced apart by approximately 2.5millimeters. The cathode wires are operated at ground potential, whereasthe anode wires 53 are operated at +3kV. The chamber defined by theinterior of the closed channel 45 is filled with an ionizable gaseousmedium such as xenon at atmospheric pressure. The cathode wires arespaced above and below the anode wires by approximately 3 millimeters.

The anode wires 53 pass through the side wall 48 of the channel 45 ingas tight sealed relation therewith and are affixed at equally spacedintervals to and along a helical delay line 56 operating at anodepotential. Opposite ends of the delay line 56 are connected torespective pulse discriminators 57 via the intermediary of pulseamplifiers 58. The outputs of the discriminators 57 are fed to atime-to-amplitude converter 59 which converts the timing betweensuccessive pulses, as derived from the discriminators 57, to a potentialproportional to the timing between such pulses. The potential output ofthe time-to-amplitude converter 59 is fed to one input of an A-to-Dconverter 61 for converting the amplitude information to a digitaloutput which is thence fed to a computer 62 which will be more fullydescribed below.

In operation, a quantum of ionizing radiation passing through the body11 passes through the window 52 and into the ionizable gas filledchamber 45. Due to the high electrical field region surrounding theindividual anode wires 53, when a quantum of ionizing radiation isabsorbed in the ionizable gas, ionization occurs which triggers anavalanche of current flow between the anode and cathode resulting in acurrent pulse on the respective anode wire 53 which is closest to theionizing event. That pulse of avalanche current is fed onto the delayline at the corresponding connection of that anode wire with the delayline 56. The pulse of current travels in opposite directions along thedelay line 56 to the ends thereof and thence via the amplifiers 58 intothe discriminators 57.

The discriminators 57 produce corresponding output pulses correspondingto the leading edge of the respective current pulses. The timedifference between successive pulses is proportional to or otherwiserepresentative of the position of the ionizing event as detected by theclosest anode wire 53. The pulses are thence fed into thetime-to-amplitude converter 59 for producing an output potentialcorresponding to the position of the ionizing event. This potential isthence converted to digital data in the analog-to-digital converter 61and fed to the computer 62. The computer stores the ionizing event in arespective channel corresponding to the position of the ionizing event.Subsequent ionizing events detected during the measurement of oneshadowgraph for each angle of θ are stored in their respective channels.Thus, the computer has stored in its memory, after one rotation of thesource through 255°, 255 sets of shadowgraph data. The computer willthen utilize these sets of shadowgraph data for reconstructing a 3-Dtomograph of the section of the body 11 under examination, as more fullydescribed below.

One of the problems with the delay line type of position sensitivedetector 21, as shown in FIGS. 7 and 8, is that it is limited to acounting rate of approximately 10⁵ counts of ionization events persecond. It is desired to employ a detector which is capable of countingat a rate of 10⁸ events per second or higher. For example, in order toobtain a 3-D tomograph having a 1/2 percent precision in the density,approximately 10⁹ counts per second are required. It is desirable thatthe 3-D tomograph data be acquired during one breath-holding period,i.e., a time of approximately 15 seconds or less. This then leads to adesired counting rate of at least 10⁸ per second.

The counting rate can be increased to at least 10⁸ per second bydeleting the delay line 56 and connecting each of the individual anodewires 53 to a respective amplifier 65 and counter 66, as shown in FIG.9. The outputs of the counters 66 are fed to an input of a multiplexer67 such that upon the completion of a shadowgraph for each angularposition of θ, the data in the counters 66 is read out via themultiplexer 67 into the computer 62 via a computer interface 68. Thecomputer 62 may comprise, for example, a PDP 11/45 minicomputer providedwith a random access memory 69 and a disc memory 71. In addition, theminicomputer 62 provides a keyboard terminal 72, and a color displayterminal 73, wherein contours of a given density in the 3-D tomographare displayed with different given colors, such that densitydifferentiation is enhanced to the human eye. In addition, theminicomputer includes a line printer 74 for printing out a 3-D densitytomograph in terms of numbers corresponding directly to density.

Referring now to FIGS. 10-19 and to the flow charts of FIG. 22 thecomputer method for reconstructing the 3-D tomographs from the sets ofangularly displaced shadowgraph data will be explained in greaterdetail. The sets of shadowgraph data, as detected by the positionsensitive detector 21, are generated by absorption of penetrationradiation by the body under analysis as taken along an array ofdivergent paths or rays. The preferred 3-D tomograph reconstructionmethod requires that the shadowgraph data correspond to absorption ofpenetrating radiation along an array of parallel paths or rays.

It has been found that the detected divergent path shadowgraph data canbe reordered into sets of shadowgraph data representative of thatobtained by arrays of parallel rays. This reordering process forreordering the divergent ray shadowgraph data into parallel rayshadowgraph data is illustrated in FIG. 10. In position θ_(i) the source13 projects a fan-shaped beam over a continuous distribution ofdivergent paths contained within the angle ψ subtended by the fan-shapedbeam. If we consider the central path or ray 75 which is identifiedr_(i),0 it is seen that this ray passes through the axis of revolution33 to the detector. Other rays denoted by r_(i),₋₃₇, r_(i),₋₃₆ . . .r_(i),₊₃₇ are spaced at 1° intervals within the fan. When the source 13of the fan-shaped beam is rotated in the positive θ direction by onedegree about the axis of rotation 33 and from the initial position ofθ_(i) to θ_(i+) 1, it will be seen that there is a new central rayidentified as r_(i) ₊₁,0 which is displaced from r_(i),0 by 1° and whichpasses through the axis of rotation 33. In addition, there is a ray r₁,1which is parallel to r_(i),0. Likewise, when the source 13 is rotated 2°to θi+2 there is a ray r_(i) ₊₂,2 parallel to both r_(i),0 and r_(i)₊₁,1. Following this rationale it will be seen that there are sets ofparallel rays according to the series indicated in FIG. 10 where r is aray or path and has subscripts θ and ψwhere θ is the angular position ofthe source 13 and ψ is the angular displacement of the ray from thecenter ray of the fan-shaped beam.

Rays may be labelled as r_(i),j where i is an index defining theposition of the source (θ_(i) = iΔθ) and j is an integer denoting theposition of each ray within a particular fan. The central ray of the fanpasses through the center of rotation and is denoted by j=0. Adjacentrays are numbered consecutively. Referring to FIG. 10 it can be seenthat it is possible to obtain at least two arrays of parallel rays asshown. If we denote a series of parallel rays inclined at θ_(i), i=0 . .. 180/Δ θ, by r'_(ij) then the reordering process can be generalized bythe following transformation where ψ fan is an odd multiple of Δθ:

    r'.sub.ij = r.sub.i.sub.+j,j   j=-j.sub.max . . . +j.sub.max

where j_(max) =(ψ fan⁻ ¹)/2Δθ. For the particular case Δθ=1° andψfan=75°, then 180 sets of parallel rays are formed with i=0, . . . ,179.

Although in the above discussion for convenience the position of thesource 13 was depicted as located at particular points, and the raysrepresented by lines, it should be understood that the source 13 anddetectors 21, etc., rotate at constant angular velocity and data isaccumulated during intervals during which the source moves continuouslyfrom one position to the next, so that θ represents a mean sourceposition during a particular interval in time. Likewise, the detector 21is sensitive to the continuous distribution of transmitted radiation sothat rays really represent the average transmission in a region ofnarrow width bounded by neighboring rays.

For relatively high resolution it is desired to obtain 180 sets ofparallel rays at one degree θ intervals. It can be shown that if 180sets of such parallel rays are to be obtained the source 13 must berotated through a total angle θ of 180° plus the fan angle ψfan. In thecase of a fan angle ψfan of 75°, the total angular displacement θ is255°. Thus, the 255 sets of divergent path or ray shadowgraph data arereordered by the computer 62 into 180 sets of parallel ray shadowgraphdata. The reordering may be accomplished by the computer 62 after thedivergent ray shadowgraphic data is stored in the respective channels ofthe memory or the data, as it is obtained at the detector 21 andmultiplexed into the memory of the computer 62, is addressed inaccordance with the desired reordering address method so that the data,as initially stored, is stored in sets of parallel shadowgraphic data.

In order to optimize the spatial resolution that can be achieved with agiven finite number of measurements it is found that the fan rays mustbe more closely spaced than the rotation step angle Δθ. If this spacingis chosen to be a fractional value of Δθ, say Δθ/n where n=2,3,4 thenthe above reordering process can still be used providing a slantapproximation is introduced. A preferred value of n is 2 yieldingΔψ=1/2° if Δθ=1°. FIG. 11 illustrates how sets of parallel rays areobtained from fan rays in this case. The slant approximation isintroduced as follows. Ray r_(i) ₊₁,1 is selected to be a member of theseries of rays parallel to r_(i),0 with spacing midway between r_(i),0and r_(i) ₊₁,2. It has been found that this approximation introduces anegligible loss of spatial resolution in the reconstruction. Two sets ofparallel rays are indicated in FIG. 11 at θ_(i) and θ_(i) ₊₁. Thus forn=2 and using the slant approximations the reordering transformationbecomes:

    r.sub.i '.sub.j = r.sub.i.sub.+j*/2, j ; j = j.sub.max, . . . j.sub.max .sup.-.sup.1,

and ##EQU1## Here j* refers to an even integer which may be either j orj+1.

Also it can be shown that the sets of reordered parallel paths or raysare not of equal lateral spacing. The spacing decreases with distanceaway from the central ray. This is depicted in FIG. 12, where the xabscissa scale represents the spacings of the reordered sets of parallelrays. The preferred 3-D method of reconstruction employs data based uponuniform lateral sapcing between parallel rays of the set. Therefore, itis desired to modify the sets of reordered parallel ray shadowgraphicdata into such data having equal lateral spacing between all parallelrays of the set.

Referring now to FIGS. 12 and 13 there is shown the method fortransforming the parallel ray shadowgraphic data into such data havingequal lateral spacing. A set of parallel rays of unequal spacing isshown at 70 in FIG. 12. In this example n=1, i.e., the slantapproximation is not used, and Δθ=1°. The x designated abscicca scaleshows the unequal lateral spacing where x_(o) =0, x₁ = Rsin 1°, x₂ =Rsin2° . . . x_(j) =Rsin jΔθ where R is the radius of the circle ofrevolution of the source 13 relative to the body 11 and j is the numberof the ray from the centray ray. The abscissa scale for equal lateralspacing of the rays 70 is that indicated by y, where y₁ =a, y₂ =2a, y₃=3a . . . y_(n) =na where ##EQU2## In the case of ψ=75° then ##EQU3##

The detected radiation intensity I₁, I₂. . . I_(n) is based uponparallel rays of unequal lateral spacing, i.e., they have the x abscissascale as shown in FIG. 13. Here the radiation intensity is assumed to beuniform within bins bounded by the midpoints between rays as shown, andthe area of bins represent the measured radiation intensities I_(i). Thex scale intensities I₁, I₂, I₃ . . . I_(n) may be transformed by arebinning process to derive parallel ray shadowgraphic intensities I'₁,I'₂ . . . I'_(n) of equal lateral spacing, as follows.

The new intensities are determined by the amount of area in old binsoverlapped by new bins. For example, ##EQU4## and, ##EQU5##

This process may be generalized by the equation: ##EQU6## where f_(ij)are the coefficients in the above equations (Eq. 1-3) and represent thefractional overlap of new bins with old bins as determined by simplegeometry as illustrated above. For speed and convenience these may becalculated in advance and stored in a disk file which may be used by thereconstruction program. Although usually the above series of equationscontain only two terms, occasionally three terms will be presentcorresponding to the case in which a new bin as illustrated in FIG. 13(y axis) is overlapped by three old bins. In the case of use of a slantapproximation with n=2,3,4, rebinning may proceed as above except thatthe values of the coordinates are given by x_(j) =Rsin jΔθ/n and##EQU7##

The coordinates of the boundaries of the old bins illustrated in FIG. 13may be calculated by an alternative method used by the computer programreferred to in the flow chart of FIG. 22. The x and y axis are taken asa line passing through the center of rotation 33 in FIG. 6, andperpendicular to a particular series of parallel rays. The new bins areequally spaced as before. The old (x axis) are defined by theintersection of the bounds of the actual fan rays with this line. Forn=2, these bounds are typically given by lines at +1/4° with respect toa perpendicular line from each particular central source position 13included in a parallel series. Typically the bins defined in this wayare not immediately adjacent as in FIG. 13 but are separated by gaps.The bins representing slant rays are introduced by using rays bounded bylines at -3/4° and -1/4° with respect to a perpendicular line from eachsource position, and will be seen to fit into the spaces between theperpendicular rays, although insignificantly small gaps will remain. Binboundaries defined in this way closely conform to those of the previousmethod, with small differences which improve the accuracy of the slantapproximation. The f_(ij) coefficients are now calculated by determiningthe amount of overlap of old bins with new bins as before.

The preferred computerized method for reconstructing the 3-D tomographsfrom the angularly displaced reordered sets of parallel ray shadowgraphsis a method disclosed in an article titled "Three DimensionalReconstruction from Radiographs and Electron Micrographs: Application ofConvolutions Instead of Fourier Transforms" appearing in the Proceedingsof the National Academy of Sciences, U.S.A., Vol. 68, No. 9, pages2236-2240 of September 1971. Briefly this method consists oftransforming the parallel ray shadowgram data into shadowgram datacorresponding to the natural logarithm 1n of the intensity of theunmodified detected radiation as a function of y, namely I' (y),normalized to the beam intensity I₀ (y). I₀ (y) is measured before theshadowgrams are made, by detecting the unabsorbed beam on each of thedetector wires 53, and applying the reordering and rebinningtransformations, and that information is stored in the computer for usein these calculations.

FIG. 14 shows a typical shadowgraph for the function 1n I'(y)/I_(O) (y)which may be referred to as g(na,θ). The linear shadowgraphs fordifferent angles θ are each scanned at intervals a and these data arethen convoluted with a function q(na) to obtain g'(na;θ) using thefollowing algorithm: ##EQU8## where ##EQU9##

The function q(na) is shown in FIG. 15 and the product of the functionof FIG. 15 with the point 78 of the shadowgraph function of FIG. 14 isshown in FIG. 16. As can be seen from the shape of the function of FIG.15, this function has zero value for even numbered intervals and dropsoff relatively quickly with interval number n so that the convolution ofthe function of FIG. 15 with that of FIG. 14 need only be evaluated at areasonably small number n of intervals a away from the point on function14 being evaluated. The individual products of the function of FIG. 15with g(na,θ) for each value of y or na of FIG. 14 are summed to derivethe function g'(na;θ) of FIG. 17 this process is known mathematically asthe convolution of g(na,θ) with q(na) as expressed in Eq. (5). In otherwords, the result of Eq. (5) is shown in FIG. 17 for a given value of θ.Thus, there is generated by the algorithm of Eq. (5) 180 sets of thefunction of FIG. 17, one for each angularly spaced set of parallel rayshadowgraph data. These 180 shadowgraphs are then back projected forcalculating the resultant 3-D reconstructed tomograph employing thefollowing algorithm: ##EQU10## where t and N are integers and r and φare the polar coordinates of the individual reconstruction matrixelements. The interval for θ is Δθ = (180/N)°, where N is the number ofshadowgraphs recorded at regular intervals over the range of -π/2 to+π/2, typically 180. In Eq. (7), the value of r cos (φ-tΔθ) will not ingeneral be a multiple of a; therefore a linear interpolation between thecalculated values of g'(na;θ) is made so that the resolution of thefinal 3-D reconstructed data obtained for f(r,φ) will depend upon thefineness of the interval a at which the shadowgraph data are availableand the consequent accuracy of the interpolation.

This method of back projection is schematically indicated in FIG. 18.More particularly, the slice of the body 11 to be examined and for whicha 3-D tomograph is to be reconstructed is considered to comprise a twodimensional matrix of elements 80 of equal size. In a typical example,the dimensions of the matrix elements are chosen equal to the spacingbetween adjacent anode wires 54, i.e., 2.5 mm. The reconstructionalgorithm Eq. (7) consists of projecting the individual values ofg'(na,θ) back across the matrix along lines perpendicular to the y axisof the particular shadowgram.

This back projection process is conveniently accomplished as follows:The coordinates r, and φ for the center of a particular matrix elementis calculated. The value of y corresponding to the point on the axis ofa particular projection intersected by a perpendicular line from thepoint r,φ is calculated. This is given by r cos (φ-tΔθ). The value ofg'(na,θ) at that value of y is calculated by means of linearinterpolation between the two values of g'(na,θ) for which na is nearesty. Hence ##EQU11## where k is the nearest integer lss than y/a. Thisprocess is repeated N times for each value of θ and the sum of eachprojected value of g(y,θ) yields the value of f(r,φ) at that grid point.The value of f(r,φ) at other grid points is computed sequentially in asimilar manner.

Referring now to FIG. 19 there is schematically indicated the problem ofposition uncertainty encountered when detecting a fan-shaped beam with arectilinear array of position sensitive detecting elements 89. Moreparticularly, as shown in FIG. 19 it is assumed that the detector 21 hassome depth d in the direction of the incoming rays 88. These rays aredivergent and in addition intercept the rectilinear array at an acuteangle. Assuming that the ray is divided into a multiplicity of detectingbins 89 it is seen that near the outer ends of the detecting array 21 agiven ray may pass through more than one bin 89. Therefore, someuncertainty is introduced relative to the position of the detected ray.

In addition, the spacing s taken along the length of the detector 21between rays of equal angular spacing ψ increases toward the outer endsof the detector 21. Thus, each bin 89 near the ends tends to detect lessradiation than bins near the center ray r_(O),O of the fan-shaped beam.Therefore, it is desirable to provide an improved position sensitivedetector which will eliminate or substantially reduce the positionaluncertainty factor and unequal spacing between rays as intercepted by arectilinear detecting array of equal spacing between detector bin 89.

Referring now to FIGS. 20 and 21, there is shown an improved positionsensitive detector 91 to replace the position sensitive detector 21 inthe embodiment of FIGS. 1 and 6. In detector 91, the detector includes agas-tight housing 92 formed by an arcuate channel structure including apair of parallel arcuate sidewalls 93 and 94, as of stainless steel,closed on the bottom by a relatively narrow arcuate end wall 95. Theopen end of the channel structure is closed by means of a high strengththin metallic foil 96, as of nickel or stainless steel, which is brazedat 97 along one marginal side edge to an inside shoulder of sidewall 93and to an arcuate rib portion 98 of side wall 94.

Opposite ends of the channel structure 92 are closed via end walls 103and 104. A removable side wall portion 90 is secured via cap screws 102to the bottom and end closing walls 95, 103 and 194 and to the arcuaterib 98 which bridges across between the end walls 103 and 104. An indiumwire seal 101 extends around the periphery of the removable cover plateportion 90 for sealing same in a gas tight manner.

The conductive housing 92 forms the cathode electrode of the detector 91and the anode electrode comprises an array of radially directed anodewires 53 centrally disposed of the chamber 92. Each anode wire 53 issupported between a pair of glass insulating terminals 105 and 106.Insulators 105 are supported from the removable cover plate 90 and theinsulators 106 are feedthrough insulators for feeding the anodepotential to the individual anode posts 107 through the envelope forconnection to the respective amplifiers 65.

The chamber 92 is filled with an ionizable gaseous medium such an xenonto a pressure above atmospheric pressure, such as 5 atmospheres. Theindividual anode wires 53, as of stainless steel, have a diameter of12.5 microns and a length of, for example, 10 centimeters. The anodewires 53 are spaced apart at 1/2° intervals of ψ with a total of 151wires 53. The projected center of the radial array of wires 53 is thesource 13 so that the individual anode wires are arrayed parallel to therays of penetrating radiation to be detected. This substantially reducesthe uncertainty and unequal spacing problems as previously alluded towith regard to rectilinear detector arrays.

For a position sensitive detector 91 based upon the concept of capturingbetween 50 and 100% of the penetrating radiation of up to 100keVincident thereon, the product of the gas fill pressure, in atmospheres,times the length of the individual anode wires 53 should equal 50atmosphere-centimeters. This means that the detector wires may be 1centimeter long if the pressure fill is at 50 atmospheres.Alternatively, the gas fill may be 5 atmospheres if the length of theindividual anode wires is 10 centimeters. Thus, the detector 91, ascontrasted with the linear detector 21, provides increased spatialresolution and improved high efficiency operation for X-ray or γ-rayenergies of 100keV and higher. Improving the spatial resolution,simplified the reconstruction of the X-ray shadowgraphic data into a 3-Dtomograph.

The flow diagram for the computer program for carrying out the 3-Dreconstruction according to the process described above with regard toFIGS. 10-18 is shown in FIG. 22 and actual computer reconstructionprograms in Fortran language, are as follows: ##SPC1##

The advantage of the fan beam penetrating ray 3-D tomograph apparatus ofthe present invention, as contrasted with prior systems utilizing bothangular rotation and transverse rectilinear translation, as exemplifiedby the aforecited U.S. Pat. No. 3,778,614, is that the lateraltranslation is eliminted and the resultant apparatus is substantiallyless complex. As a result, the time required to obtain the amount ofshadowgraphic data required for high resolution, i.e., 1% accuracy 3-Dreconstruction, is reduced to times less than a breath-holding period sothat portions of the body subject to movement with breathing and thelike can be obtained without blurring due to body movement. For example,the present invention permits 3-D tomographs to be obtained of the lungswithout blurring due to movement.

What is claimed is:
 1. In a method of penetrating ray 3-D tomography thesteps of:directing a divergent beam of penetrating radiation through abody to be examined from a source on one side of the body to a detectoron the other side of the body; effecting relative angular displacementbetween the divergent beam of penetrating radiation and the body;detecting the penetrating radiation that has passed through the body ata number of angularly spaced positions within the angle subtended by thedivergent beam as a function of the angular position of the divergentbeam to derive sets of detected radiation data representative of aplurality of angularly spaced shadowgrams of absorption or transmissionof the penetrating radiation by the body, each of said shadowgramsrepresenting the transmission of the penetrating radiation through thebody along an array of divergent paths subtended by the divergent beam,and different ones of said sets of angularly spaced shadowgraphic datacorresponding to different sets of intersecting rays of penetratingradiation; and reordering the sets of data corresponding to absorptionor transmission shadowgraphic data of divergent rays of said penetratingradiation into sets of data corresponding to absorption or transmissionshadowgrams of parallel rays of said penetrating radiation.
 2. Themethod of claim 1 including the step of, reconstructing a 3-D tomographfrom said sets of parallel ray shadowgrams.
 3. The method of claim 1including the step of, transforming said sets of data corresponding toabsorption or transmission shadowgrams into sets of logarithmicshadowgraphic data corresponding to the natural logarithm of saidshadowgraphic data normalized to the beam intensity.
 4. The method ofclaim 1 wherein the step of detecting the penetrating radiation toderive sets of angularly spaced shadowgraphic data comprises detectingradiation passing through the body at a number of angularly spacedpositions within the angle subtended by the divergent beam, such angularspacing within the divergent beam being smaller in angular spacing thanthe angular spacing between adjacent shadowgrams.
 5. The method of claim4 wherein the detection angular spacing within the divergent beam iswithin the range of one quarter to three quarters of the angular spacingbetween adjacent shadowgrams.
 6. The method of claim 4 wherein thedetection angular spacing within the divergent beam is one half of theangular spacing between adjacent shadowgraphs.
 7. The method of claim 1wherein the step of reordering the sets of divergent ray data into setsof parallel ray data includes the step of rebining the parallel ray datawhich consists of unequally laterally spaced sets of parallel ray datainto equivalent sets of equally laterally spaced parallel ray data sothat the reordered and rebined parallel ray shadowgram data correspondsto sets of parallel rays of generally equal lateral spacing.
 8. Themethod of claim 3 including the step of convoluting the naturallogarithmic shadowgraphic data with the function q(na) to obtain g'(na;θ) according to the algorithm: ##EQU12##where ##EQU13##where θ is theangle of rotation of the source about the center of rotation, a is thebin width for equally spaced bins, n is an integer variable of theequation that corresponds to the particular bin number, p is an integerindex, and g is the natural logarithmic shadowgraphic data.
 9. Themethod of claim 8 including the step of reconstructing a penetrating ray3-D tomograph of the body according to the algorithm: ##EQU14##where tand N are integer index numbers; r and φ are the polar coordinates ofthe individual reconstruction matrix elements, the interval for θ is θ₀(180/N')° where N' is the number of shadowgraphs recorded at regularangular intervals of θ over the range of -π/2 to +π/2; and f is the 3-Dtomograph.
 10. The method of claim 1 wherein the step of effectingrelative angular displacement between the divergent beam of penetratingradiation and the body comprises effecting said relative angulardisplacement in a manner which is substantially free of relative lateraltranslation therebetween.
 11. In an apparatus for obtaining a 3-Dtomograph of a body to be examined:means for directing a divergent beamof penetrating radiation through the body to be examined from a sourceon one side of the body to a detector on the other side of the body;means for effecting relative angular displacement between the divergentbeam of penetrating radiation and the body; means for detecting thedivergent penetrating radiation that is passed through the body at anumber of angularly spaced positions within the angle subtended by thedivergent beam as a function of the relative angular position of thedivergent beam relative to the body to derive sets of detected radiationdata representative of sets of angularly spaced divergent rayshadowgrams of absorption or transmission of the divergent penetratingradiation by the body with different ones of said angularly spaced setsof the divergent ray shadowgram data corresponding to different sets ofintersecting rays of divergent penetrating radiation; and means forreordering the sets of data corresponding to absorption or transmissionshadowgrams of divergent rays of said penetrating radiation into sets ofdata corresponding to absorption or transmission shadowgrams of sets ofparallel rays of said penetrating radiation.
 12. The apparatus of claim11 including, means for reconstructing a 3-D tomograph from said sets ofparallel ray shadowgrams.
 13. The apparatus of claim 11 including, meansfor transforming one of said sets of shadowgram data into sets oflogarithmic shadowgraphic data corresponding to the natural logarithm ofsaid parallel ray shadowgraphic data normalized to the beam intensity.14. The apparatus of claim 11 wherein said detecting means for detectingthe penetrating radiation to derive sets of angularly spacedshadowgraphic data comprises, means for detecting said penetratingradiation passing through the body at a number of angularly spacedpositions within the angle subtended by the divergent beam, such angularspacing of said detecting means within the divergent beam being smallerin angular spacing than the angular spacing between adjacentshadowgrams.
 15. The apparatus of claim 14 wherein said angular spacingof said detecting means within the divergent beam is within the range ofone quarter to three quarters of the angular spacing between adjacentshadowgrams.
 16. The apparatus of claim 14 wherein the angular spacingof said detecting means within the divergent beam is one half of theangular spacing between adjacent shadowgrams.
 17. The apparatus of claim11 wherein said means for reordering the sets of divergent ray data intosets of parallel ray data includes, means for re-bining the parallel raydata which consists of unequally laterally spaced sets of parallel raydata into equivalent sets of equally laterally spaced parallel ray dataso that the reordered and re-bined parallel ray shadowgram datacorresponds to sets of parallel rays of generally equal lateral spacing.18. The apparatus of claim 13 including, means for convoluting thenatural logarithmic shadowgraphic data with the function of q(na) toobtain g'(na;θ) approximatey according to the algorithm; ##EQU15##where##EQU16##where θ is the angle of rotation of the source about the centerof rotation, a is the bin width for equally spaced bins, n is an integervariable of the equation that corresponds to the particular bin number,p is an integer index and g is the natural logarithmic shadowgraphicdata.
 19. The apparatus of claim 18 including, means for reconstructinga penetrating ray 3-D tomograph of the body according to the algorithm:##EQU17##where t and N are integer index numbers; r and φ are the polarcoordinates of the individual reconstruction matrix elements, theinterval for θ is, θ₀ (180/N')° where N' is the number of shadowgraphsrecorded at regular angular intervals of θ; and f is the 3-D tomograph.20. The apparatus of claim 11 wherein said means for effecting relativeangular displacement between the divergent beam of penetrating radiationand the body includes means for effecting said relative angulardisplacement in a manner which is substantially free of relative lateraltranslation therebetween.